Geometric correspondence for ensembles of nonregular shapes

Abstract

An ensemble of biological shapes can be represented and analyzed with a dense set of point correspondences. In previous work, optimal point placement was determined by optimizing an information theoretic criterion that depends on relative spatial locations on different shapes combined with pairwise Euclidean distances between nearby points on the same shape. These choices have prevented such methods from effectively characterizing shapes with complex geometry such as thin or highly curved features. This paper extends previous methods for automatic shape correspondence by taking into account the underlying geometry of individual shapes. This is done by replacing the Euclidean distance for intrashape pairwise particle interactions by the geodesic distance. A novel set of numerical techniques for fast distance computations on curved surfaces is used to extract these distances. In addition, we introduce an intershape penalty term that incorporates surface normal information to achieve better particle correspondences near sharp features. Finally, we demonstrate this new method on synthetic and biological datasets.

Fig. 1

Points near sharp features (left) are not able to achieve good distributions with Euclidean distance, because they do not lie in the same tangent space, which is necessary for movement that is constrained to the surfaces. Points may be nearby and interact (center) even though they sample very different parts of the surface. Points on nearby features (right) on different shapes (blue and green) can come into incorrect correspondence if the system considers only distance.

Fig. 2

(a) An example of a triangle mesh used for geodesic distance computations. (b) Configuration for two-layered interpolation of geodesic distance between arbitrary points: x and y are contained in triangles defined by vertices (x₁, x₂, x₃) and (y₁, y₂, y₃) respectively. The geodesic distances between vertices for all shapes are precomputed on a GPU.

Fig. 3

Mean shape computed from the proposed method (left) and the original method (right), projected onto the first (top) and second (bottom) PCA modes, and ±2 standard deviations

Fig. 4

Visualizing mean differences between normal and ischemic groups (blue denotes expansion and yellow denotes contraction) using [1] (top row) and the proposed method (bottom row)